From: "R. Bryant" <b1r2y3a4n5t@math.duke.edu>
Subject: Re: Spinors
Date: Tue, 02 Jan 2001 13:55:52 -0500
Newsgroups: sci.math.research
Summary: Classifications of orbits of actions of Spin(n)

John Baez wrote:

> A couple of questions about spinors.
>
> Is there a nice description of the orbits of the action of Spin(n) on
> its spinor representations?  I'm mostly interested in real spinors....
> I know, for example, that Spin(9) acts transitively on the unit sphere
> of its spinor representation on R^16.  Also, since the dimension of the
> spinor reps grows exponentially, while that of Spin(n) grows quadratically,
> eventually Spin(n) can no longer act transitively on the unit sphere of
> its spinor representation.  But when does this happen?  And what are the
> orbits like then?

I don't know whether there is any comprehensive description of
the orbits of Spin(n) on its lowest dimensional faithful representation S
for all n.  I do know that it never acts transitively on S when n > 9.  This
follows from the the classification by Borel in 1950 of the subgroups
of the orthogonal groups that act transitively on spheres. (See  Borel, Armand

Le plan projectif des octaves et les sph�ères comme espaces homog�ènes.
C. R. Acad. Sci. Paris 230, (1950). 1378--1380.)

I do know that, when n = 10, the generic orbit of Spin(10) acting on S =
R^{32}
has dimension 30, i.e., is a hypersurface in the 31-sphere, and the stabilizer

of a point on such an orbit is isomorphic to SU(4).  There are two
smaller orbits, one of dimension 21 (the stabilizer in Spin(10) is SU(5))
and the other of dimension 24 (the stabilizer in Spin(10) is Spin(7)).  What
happens in higher dimensions, I don't know.

> Also, in what year was triality first discovered?

I can't say for certain, but I believe that the first use of the term
'triality'
to describe the relations among the three 8 dimensional representations
of Spin(8) was in E. Cartan's 1925 paper, 'Le principe de dualite et la
theorie
des groupes simple et semi-simples'.   Of course, you can argue that this
relationship was discovered by Cartan even earlier in his fundamental paper
of  1914, 'Les groupes reels, simples, finis et continus', since he explicitly

remarks, at the end of Part IV, that , in the case of D_4, there are three
systems of weights can can be taken as fundamental, but that they are all
conjugate by outer automorphisms.   However, I think that this is stretching
a bit.

    Yours,

    Robert Bryant